In this lecture Professor Poppitz derived a rest frame solution of the Dirac equation, then demonstrated that the generalization to non-zero momentum satisfied the equation. We also saw that Dirac spinor solutions of the Dirac equation are KG equation solutions, and touched on the relation of some solutions to the helicity operator.

This post doesn’t have a web version, since my latex -> wordpress-mathjax script doesn’t have support for the theorem/lemma environments that I used for Monday’s notes, and I don’t have time to implement that right now.

These also include a portion of the notes from Lecture 18 (not yet posted), as it made sense to group all the Pauli matrix related content. This particular set of notes diverges from the format presented in class, as it made sense to me to group things in this particular lecture in a more structured definition, theorem, proof style. I’ve added a number of additional details that I found helpful, as well as a couple of extra problems (some set as formal problems at the end, and others set as theorem or lemmas in with the rest.)

In class yesterday (lecture 19, notes not yet posted) we used \( \Bsigma^\T = -\sigma_2 \Bsigma \sigma_2 \), which implicitly shows that \( (\Bsigma \cdot \Bx)^\T \) is a reflection about the y-axis.
This form of reflection will be familiar to a student of geometric algebra (see [1] — a great book, one copy of which is in the physics library). I can’t recall any mention of the geometrical reflection identity from when I took QM. It’s a fun exercise to demonstrate the reflection identity when constrained to the Pauli matrix notation.

When we expand \( (\Bsigma \cdot \Bx)^\T \) and find
\begin{equation}\label{eqn:reflection:n}
(\Bsigma \cdot \Bx)^\T
=
\sigma^1 x^1 – \sigma^2 x^2 + \sigma^3 x^3,
\end{equation}
it is clear that this coordinate expansion is a reflection about the y-axis. Knowing the reflection formula above provides a rationale for why we might want to write this in the compact form \( -\sigma^2 (\Bsigma \cdot \Bx) \sigma^2 \), which might not be obvious otherwise.

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Here are my evaluation comments for QFT I. The university provides an anonymous facility to submit course feedback, but since I have no conflicts that require anonymization, I’m posting my commentary (and rationale for some of my list selections) publicly.

Q) Please comment on the overall quality of the instruction in this course.

Professor Poppitz’s knowledge of the subject matter is impressive and thorough. I expect that this is a particularly difficult course to teach and think that he has done an admirable job trying to work through the maximum amount of material in the limited time available in this course.

It is challenging but fun game (albeit a slightly masochistic one) to keep up with Prof Poppitz’s blistering pace through the course material. Poppitz often says “Phew!!” at the end of the race to complete a long derivation in the allotted time, and I’ll be saying the same thing at the end of this course.

The barrage of abstract material covered in a lecture is often sufficient to leave me with a headache, and it takes a few hours to recover from each class. It takes a few more hours after that to digest the material at a human pace.

This course would strongly benefit from video recorded lectures that some of the more educationally progressive academic institutions currently provide (i.e. MIT’s OCW and Yale’s “Open Yale” courses). With the exception of the UofT SciNet group (phy1610 — Scientific computing for Physicists), I’m not aware of any UofT physics courses that provide such recordings. For phy2403, video recordings would be particularly valuable, as it would allow the student to “pause” the Professor and work through the material presented at an individually suitable pace.

Q) Please comment on any assistance that was available to support your learning in the course.

Prof Poppitz was available continually on the course forum, after class briefly, and in weekly office hours. I received a great deal of helpful assistance from him during the course.

Q) Compared to other courses, the workload for this course was…

I picked Heavy (not Very Heavy), but I’m not in a good position to evaluate since I’m only taking one course.

Q) I would recommend this course to other students.

I picked Mostly (not Strongly). I wouldn’t recommend this course to anybody who was not adequately prepared. I’m not sure that I was. It is a very tough course. I was continually impressed with the other students in the class. I’ve worked slowly for years to gradually build up the background required to take this course, and all the rest of these younglings are downing the material with seeming ease. There are a lot of exceptionally smart students enrolled on this course.

DISCLAIMER: Very rough notes from class (today VERY VERY rough).

These are notes for the UofT course PHY2403H, Quantum Field Theory, taught by Prof. Erich Poppitz, fall 2018.

Review: S-matrix

We defined an \( S-\)matrix
\begin{equation}\label{eqn:qftLecture17:20}
\bra{f} S \ket{i} = S_{fi} = \lr{ 2 \pi }^4 \delta^{(4)} \lr{ \sum \lr{p_i – \sum_{p_f} } } i M_{fi},
\end{equation}
where
\begin{equation}\label{eqn:qftLecture17:40}
i M_{fi} = \sum \text{ all connected amputated Feynman diagrams }.
\end{equation}
The matrix element \( \bra{f} S \ket{i} \) is the amplitude of the transition from the initial to the final state. In general this can get very complicated, as the number of terms grows factorially with the order.

We also talked about decays.

Scattering in a scalar theory

Suppose that we have a scalar theory with a light field \( \Phi, M \) and a heavy field \( \varphi, m \), where \( m > 2 M \). Perhaps we have an interaction with a \( z^2 \) symmetry so that the interaction potential is quadratic in \( \Phi \)
\begin{equation}\label{eqn:qftLecture17:60}
V_{\text{int}} = \mu \varphi \Phi \Phi.
\end{equation}
We may have \( \Phi \Phi \rightarrow \Phi \Phi \) scattering.

We will denote diagrams using a double line for \( \phi \) and a single line for \( \Phi \), as sketched in

and can have additional virtual particles created, with diagrams like fig. 4.

fig. 4. More virtual particles.

We will see (QFT II) that this leads to an addition imaginary \( i \Gamma \) term in the propagator
\begin{equation}\label{eqn:qftLecture17:160}
\frac{i}{s – m^2 + i \epsilon}
\rightarrow
\frac{i}{s – m^2 – i m \Gamma + i \epsilon}.
\end{equation}
If we choose to zoom into the such a figure, as sketched in fig. 5, we find that it contains the interaction of interest for our diagram, so we can (looking forward to currently unknown material) know that our diagram also has such an imaginary \( i \Gamma \) term in its propagator.

fig. 5. Zooming into the diagram for a higher order virtual particle creation event.

Decay rates.

We have an initial state
\begin{equation}\label{eqn:qftLecture17:240}
\ket{i} = \ket{k},
\end{equation}
and final state
\begin{equation}\label{eqn:qftLecture17:260}
\ket{f} = \ket{p_1^f, p_2^f \cdots p_n^f}.
\end{equation}
We defined decay rate as the ratio of the number of initial particles to the number of final particles.

Conservation of the field momentum.

It turns out that examining the reasons that we can say that the field momentum is conserved also sheds some light on the question. \( P^i \) is not an a-priori conserved quantity, but we may use the charge conservation argument to justify this despite it not having a four-vector nature (i.e. with zero four divergence.)

The momentum \( P^i \) that we have defined is related to the conserved quantity \( T^{0\mu} \), the energy-momentum tensor, which satisfies \( 0 = \partial_\mu T^{0\mu} \) by Noether’s theorem (this was the conserved quantity associated with a spacetime translation.)

Vanishing for large \( \Bp \) doesn’t help the whole integral vanish, but we can resort to the Riemann-Lebesque lemma [1] instead and interpret this integral as one with a plain old high frequency oscillation that is presumed to vanish (i.e. the rest is well behaved enough that it can be labelled as \( L_1 \) integrable.)

We see that only the non-time dependent portion of \( \mathbf{P} \) matters from a conserved quantity point of view, and having killed off all the time dependent terms, we are left with a conservation relationship for the momenta \( \spacegrad \cdot \BP = 0 \), where \( \BP \) in normal order is just
\begin{equation}\label{eqn:momentum:200}
: \BP : = \int \frac{d^3 p}{(2 \pi)^3} \Bp a_\Bp^\dagger a_\Bp.
\end{equation}